Related papers: The GW/PT conjectures for toric pairs

We write the equivariant Todd class of a general complete toric variety as an explicit combination of the orbit closures, the coefficients being analytic functions on the Lie algebra of the torus which satisfy Danilov's requirement.

We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of the complex projective line with orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of…

The Welschinger numbers, a kind of a real analog of the Gromov-Witten numbers which count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any real generic…

We study rational curves of degree two on a smooth sextic 4-fold and their counting invariant defined using Donaldson-Thomas theory of Calabi-Yau 4-folds. By comparing it with the corresponding Gromov-Witten invariant, we verify a…

In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the…

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic…

We prove that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, provided that the decomposition is considered over a…

We apply a theorem of Gel'fand, Goresky, MacPherson, and Serganova about matroid polytopes to study semistability of partial flags relative to a T-linearized ample line bundle of a flag space F = SL(n)/P where T is a maximal torus in SL(n)…

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank $r$ on $X$ in terms of…

In this note we prove that the crepant transformation conjecture for a crepant birational transformation of Lawrence toric DM stacks studied in \cite{CIJ} implies the monodromy conjecture for the associated wall crossing of the symplectic…

Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient.…

Donaldson-Thomas (DT) invariants of a quiver with potential can be expressed in terms of simpler attractor DT invariants by a universal formula. The coefficients in this formula are calculated combinatorially using attractor flow trees. In…

We investigate the effect of a general toric wall crossing on genus zero Gromov-Witten theory. Given two complete toric orbifolds $X_+$ and $X_-$ related by wall crossing under variation of GIT, we prove that their respective $I$-functions…

We discuss as a fundamental characteristic of orthogonal polynomials like the existence of a Lie algebra behind them, can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we put…

Graph polytopes arising from vertex-weighted graphs were first introduced by B\'ona, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is…

The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants)…

Kenyon and Wilson showed how to test if a circular planar electrical network with $n$ nodes is well-connected by checking the positivity of $\binom{n}{2}$ central minors of the response matrix. Their test is based on the fact that any…

We provide an enumerative meaning of the mirror maps for toric Calabi-Yau orbifolds in terms of relative Gromov-Witten invariants of the toric compactifications. As a consequence, we obtain an equality between relative Gromov-Witten…

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials,…